Compressive sensing is an approach to sensing signals that enables sampling signals at below the Nyquist rate. With compressive sampling, it is possible to reconstruct a sampled signal perfectly, even though the signal is sampled at below the Nyquist rate. One approach to compressive sensing is to create a time encoded representation of a signal. This is done by first converting an analog signal that has a continuous voltage range to a binary amplitude signal. Then the zero crossing time points are used to form a time encoded signal representation.
Time-encoded-based compressive sensing (TE-CS) system has been used in the prior art. Examples of time encoders for compressive sensing are to be found in U.S. Ser. No. 12/262,691, “Compressed Sensing Time Encoded Based Analog-to-Digital Converter”, filed Oct. 31, 2008 by P. Petre and J. Cruz-Albrecht, U.S. Pat. No. 7,403,144 “Pulse Domain Encoders and Filter Circuits” to J. Cruz-Albrecht and P. Petre, and U.S. Pat. No. 7,515,084 “Analog-to-Digital Converter using Asynchronous Pulse Technology” to J. Cruz-Albrecht, P. Petre and J. Jensen.
Other prior art references that discuss compressive sensing and/or time encoding include: David L. Donoho, “Compressed Sensing”, IEEE Trans. on Information Theory, Vol. 52, No. 4, pp. 1289-1306, April 2006; E. Candes and T. Tao, “Decoding by linear programming”, IEEE Trans. Information Theory, Vol. 51, 2005, pp 4203-4215; E. Candes and T. Tao, “Near Optimal Signal Recovery from Random Projections: University Encoding Strategies?”, IEEE Trans. on Information Theory, Vol. 52, No. 12, pp. 5406-5425, December 2006; and Aurel A. Lazar and László T. Tóth, “Perfect Recovery and Sensitivity Analysis of Time Encoded Bandlimited Signals”, IEEE Trans. on Circuits and Systems-I: Regular Papers, Vol. 51, No. 10, October 2004, pp. 2060-2073.
Time encoding is desirable because the performance of current analog to digital converters (ADCs) is not adequate for direct digitization of wide bandwidth and high dynamic range RF signals. For example, a 10 GHz bandwidth state-of-the-art ADC can provide only 5-bit resolution. To accomplish the goal of overcoming the limitation of the current ADCs, one technique called Analog Pulse Processing (APP) or Time Encoding Machine (TEM) represents a continuous analog signal with an asynchronous time-sequence with known binary amplitude. The asynchronous system overcomes many limitations found with synchronous ADCs. Hence it can successfully encode the signal with high bandwidth.
In general, a signal can be perfectly reconstructed from time sequences if the Nyquist rate is satisfied, i.e., if the number of time points within a period is larger than the Shannon number. Moreover, if the signal is sparse in a certain domain, according to compressive sensing theory, the signal can still be fully reconstructed with high probability even if the number of time points is less than the Shannon number. Combining TEM and compressive sensing can enable sampling and reconstructing RF signals of wide bandwidth and high dynamic range.
However, in prior art time encoding machines (TEMs) the measurements may not be “random” enough to comply with compressive sensing theory, which results in an inability to properly reconstruct a sampled signal. In particular, a sampled signal may not be properly reconstructed using compressive sensing in these prior art time encoding implementations, because these prior time encoding processes are signal dependent and the time encoded signal may not be “random” enough for compressive sensing theory to apply.
What is needed is a time encoding machine that can be used successfully for compressive sensing even if the signal is not “random” enough. Also needed are improved methods of reconstructing a sampled signal. The embodiments of the present disclosure answer these and other needs.